What is the concept of channels in CNNs?
Say you have a colored image that is 200x200 pixels. The standard is such that the input matrix is a 200x200 matrix with 3 channels. The first convolutional layer would have a filter that is size $N×M×3$, where $N,M&lt;200$ (I think they're usually set to 3 or 5). 
  
  Would it be possible to structure the input data differently, such that the number of channels now becomes the width or height of the image? i.e., the number of channels would be 200, the input matrix would then be 200x3 or 3x200. What would be the advantage/disadvantage of this formulation versus the standard (# of channels = 3)? Obviously, this would limit your filter's spatial size, but dramatically increase it in the depth direction. 


The different dimensions (width, height, and number of channels) do have different meanings, the intuition behind them is different, and this is important. You could rearrange your data, but if you then plug it into an implementation of CNNs that expects data in the original format, it would likely perform poorly.

The important observation to make is that the intuition behind CNNs is that they encode the "prior" assumption or knowledge, the "heuristic", the "rule of thumb", of location invariance. The intuition is that, when looking at images, we often want our Neural Network to be able to consistently (in the same way) detect features (maybe low-level features such as edges, corners, or maybe high-level features such as complete faces) regardless of where they are. It should not matter whether a face is located in the top-left corner or the bottom-right corner of an image, detecting that it is there should still be performed in the same way (i.e. likely requires exactly the same combination of learned weights in our network). That is what we mean with location invariance.

That intution of location invariance is implemented by using "filters" or "feature detectors" that we "slide" along the entire image. These are the things you mentioned having dimensionality $N \times M \times 3$. The intuition of location invariance is implemented by taking the exact same filter, and re-applying it in different locations of the image.

If you change the order in which you present your data, you will break this property of location invariance. Instead, you will replace it with a rather strange property of... for example, "width-colour" invariance. You might get a filter that can detect the same type of feature regardless its $x$-coordinate in an image, and regarldess of the colour in which it was drawn, but the $y$-coordinate will suddenly become relevant; your filter may be able to detect edges of any colour in the bottom of an image, but fail to recognize the same edges in the top-side of an image. This is not an intuition that I would expect to work successfully in most image recognition tasks.



Note that there may also be advantages in terms of computation time in having the data ordered in a certain way, depending on what calculations you're going to perform using that data afterwards (typically lots of matrix multiplications). It is best to have the data stored in RAM in such a way that the inner-most loops of algorithms using the data (matrix multiplication) access the data sequentially, in the same order that it is stored in. This is the most efficient way in which to access data from RAM, and will result in the fastest computations. You can generally safely expect that implementations in large frameworks like Tensorflow and PyTorch will already require you to supply data in whatever format is the most efficient by default.